Question: Let $z$ be a complex number such that
\[z^2 + |z|^2 = 3 - 5i.\]Find $|z|^2.$
Let $z = a + bi,$ where $a$ and $b$ are real numbers.  Then $z^2 = (a + bi)^2 = a^2 + 2abi - b^2$ and $|z|^2 = a^2 + b^2,$ so
\[a^2 + 2abi - b^2 + a^2 + b^2 = 3 - 5i.\]Equating real and imaginary parts, we get
\begin{align*}
2a^2 &= 3, \\
2ab &= -5.
\end{align*}From the first equation, $a^2 = \frac{3}{2}.$  From the second equation,
\[b = -\frac{5}{2a},\]so
\[b^2 = \frac{25}{4a^2} = \frac{25}{4 \cdot 3/2} = \frac{25}{6}.\]Therefore,
\[|z|^2 = a^2 + b^2 = \frac{3}{2} + \frac{25}{6} = \boxed{\frac{17}{3}}.\]